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Descartes' rule of signs. In mathematics, Descartes' rule of signs, described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting zero coefficients ...
Descartes' rule of signs asserts that the difference between the number of sign variations in the sequence of the coefficients of a polynomial and the number of its positive real roots is a nonnegative even integer. It results that if this number of sign variations is zero, then the polynomial does not have any positive real roots, and, if this ...
All results described in this article are based on Descartes' rule of signs. If p(x) is a univariate polynomial with real coefficients, let us denote by # + (p) the number of its positive real roots, counted with their multiplicity, [1] and by v(p) the number of sign variations in the sequence of its coefficients.
If it is not the case, zero is a root, and the localization of the other roots may be studied by dividing the polynomial by a power of the indeterminate, getting a polynomial with a nonzero constant term. For k = 0 and k = n, Descartes' rule of signs shows that the polynomial has exactly one
If there are two or more sign variations Descartes' rule of signs implies that there may be zero, one or more real roots inside the interval (0, ∞); in this case consider separately the roots of p(x) that lie inside the interval (0, 1) from those inside the interval (1, ∞). A special test must be made for 1.
Root-finding algorithm. In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the zeros of a function cannot ...
Sturm's theorem provides a way for isolating real roots that is less efficient (for polynomials with integer coefficients) than other methods involving Descartes' rule of signs. However, it remains useful in some circumstances, mainly for theoretical purposes, for example for algorithms of real algebraic geometry that involve infinitesimals. [3]
Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution x = p ⁄ q, written in lowest terms so that p and q are relatively prime, satisfies: p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n.